© 2019 American Physical Society. The validity of Fourier's law in telescopic nanowires is tested by means of molecular dynamics simulations. We observe that the radial dependence of the heat-flux profile, the temperature jump, and the appearance of vorticity obtained in molecular dynamics telescopic wires near the contact point acquire a hydrodynamic character, and we show that they are incompatible with Fourier's law. We propose the Guyer-Krumhansl equation as a generalization capable of capturing these hydrodynamic effects. Lattice-dynamics results show that the thermal average of the confined modes shows no radial dependence of the vibrating energy inside the wire. This means that hydrodynamic effects could not be related to the confinement effects in these small systems.